Most likely he remained in Middle-earth during the Fourth Age and continued his work with the flora and fauna of Southern Mirkwood, undisturbed at last by distractions such as the schemes of the White Council and the War of the Ring. Although Radagast was considered to have 'failed' in his mission, he was sent at the prayer of Yavanna and so it is possible that he actually did complete his mission. In any case, I believe that Radagast possessed a good heart - even if he was reluctant to become involved in the affairs of the wider world - and he would probably have done all he could to aid those who sought him out in the Fourth Age (after a bit of grumbling).